New lattice point asymptotics on products of upper half-planes

BRUGGEMAN, R. W.; GRUNEWALD F.; MIATELLO R.J.

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2011 vol. 2011 p. 1510 - 1519

1073-7928

Sadly, Fritz Grunewald died when this work was in the process of revision. We dedicate this paper to his memory. Let Γ be an irreducible lattice in PSL2(ℝ)d (d ∈ ℕ) and z a point in the d-fold direct product of the upper half-plane. We study the discrete set of componentwise distances D(Γ, z) ⊂ ℝd defined in (2). We prove asymptotic results on the number of γ ∈Γ such that dist(z, γz) is contained in strips expanding in some directions and also in expanding hypercubes. The results improve the existing error terms, [6], and generalize the best known error term for d = 1, due to Selberg.